I looked at the catcher's ERA difference between my two drafted catchers (2000 Mitch Meluskey and 1977 Cliff Johnson) as well as that of my Triple-A catcher (2017 Kurt Suzuki) and saw that the C-ERA for Suzuki was more than a run lower than Meluskey and two runs lower than Johnson. Defensively, Suzuki is A/A+/D, Meluskey is C/A/D and Johnson is D-/C/C with catcher a secondary position for him.

Is the C-ERA built into the catcher's defensive rating, or is there another metric that rates it, or is it just random and not affecting my pitchers at all except luck?

catcher's era won't have any impact on your pitchers or anything else